\begin{answer}

    Note that 

    $$
    \begin{aligned}
        D_{KL}(P\|Q)  &= E_{x \sim P(x)}(\log \frac{P(x)}{Q(x)}) \\
        &= -E_{x\sim P(x)}(\log\frac{Q(x)}{P(x)})\\
        &\ge -\log E_{x\sim P(X)}(\frac{Q(x)}{P(x)})    = -\log 1 = 0
    \end{aligned}
    $$

    The inequality becomes equality only when $Q(x) / P(x) = c$ is a constant. And this implies that $Q(x) = P(x)$.

\end{answer}
